Re: Countable models of ZFC



Rupert wrote:
herbzet wrote:
Rupert wrote:
On Oct 3, 9:34 pm, george <gree...@xxxxxxxxxx> wrote:
On Sep 26, 5:35 am, Rupert <rupertmccal...@xxxxxxxxx> wrote:

You no doubt have an idea about what _you_ mean by set. ZFC
has _its_ idea of what a set is -- or rather, ideas. If
ZFC is coherent, there are _different things_ that answer to
ZFC's description of what a set is.

Whenever we are doing mathematics, I suppose we could always stop and
ask ourselves "Hey, I wonder which model of ZFC we're working in? I
wonder what it all means?" But I don't see why there is special cause
to do so on this occasion and not on other occasions.

Yes. I don't know that this is a special case. The technicalities of
what might be a "standard" model are somewhat beyond me. No doubt
your context is just discussing results within the framework of
ordinary model theory.

What is interesting to me is the problem of providing a model
of the language of set theory in terms of sets. How can we
model the term "set" by calling on a set as our model?

What is the purpose of a model of a theory? One reason
is to provide a semantics for the theory in which all the
theorems come out true, and so demonstrate the consistency
of the theory.

If we want a semantics for set theory, how is it meaningful
to use sets as the referents of the terms? Seems circular.

On the other hand, if we regard it as meaningful to use
sets to provide a semantics for other theories, why would
we object to using sets to provide a semantics for the theory
of sets? Why is providing a set-theoretic model of some
theory considered meaningful at all? Surely because it
intrinsically makes some kind of simple sense.

Of course, these may be the wrong questions, since we cannot
provide a set-theoretic model of set theory in the first place.


I don't like talking about proper classes.
When I say "M is a set", I don't mean it's a set living in some model,
I just mean it's a set, period.

You sound like MoeBlee! _I_ think that you just have an intended
model in mind. Either that or you have no position whatever on the
"truth" of undecideable propositions of ZFC.

Some people might want to use a theory which allows proper classes and
then the universe counts as a model too. Of course, this model is
standard (just as the standard kilogram weighs one kilogram).

By all means, let's restrict the discussion to ZFC if that's what
you're comfortable with. ZFC is strong enough that we're not really
sacrificing anything thereby.

You have to be able to make sense of "x is a member of y" all by
itself,

It is the axioms that makes sense of "x e y". Unfortunately, perhaps,
they leave room for different interpretations.

You're saying you don't understand the first-order language of set
theory, you don't know what the intended model is.

A nice quote I ran into recently:

'Hence, a mathematician who states that he "believes an axiom
to be true" is actually indicating what he considers to be the
principal interpretation of the theory."

(from http://www.friesian.com/goedel/chap-1.htm#sect-5 )

Fine, then you
don't understand mathematics. You can just interpret mathematics
formalistically.

In that case I don't need models at all. All I'm interested
in is what strings can be produced from other strings.

But even then, models are handy for independence proofs.

This doesn't have a bearing on what I said from the
mathematical point of view. What I said was perfectly good
mathematics, whether you're a formalist or a realist.

There is no doubt that the intended interpretation is what we
ordinarily mean in natural language by the words "is a member of".
It is assumed that the axioms capture this intended meaning,
among others.

Obviously the meaning of the basic terms has to come before the
axioms, not the other way around.

Now that's an interesting proposition. Maybe I'll come back to
that later.

If we don't understand the basic
terms, we don't understand the axioms and they can't tell us anything.
Either we understand the basic terms or we don't. The fact that the
axioms have many different models doesn't matter.

Then I guess the existence of undecidable propositions doesn't matter.

not just "x is a member of y in such-and-such a model". There
has to be such a notion available, or your semantics will never get
off the ground.

IF THAT model was nonstandard then merely restricting its membership
relation to some submodel IS NOT going to guarantee that that submodel
(in this case, (M,E)) is standard. SOME nonstandard models DO have
nonstandard submodels.

If you're going to take the view that you can never understand any
sentence unless you've specified what model it's relativized to,
you're going to tie yourself in knots.

The truth of some sentences of the language of ZFC are relative to
a model. There's no getting around that.


Either you understand the first-order language of set theory or you
don't. If you understand it, then you don't need to engage in all of
George's ranting about "outer models".

I wouldn't be inclined to call the intended model a model, because I
don't like talking about models that are proper classes. But never
mind that. We might sometimes want to speak of an intended model
(although on this occasion it adds nothing but unnecessary confusion).

Ah. You distinguish between a standard model and an intended model.

But are you saying we can't know what the intended model is,

No. That is not my position. My position is that I don't
understand the logic of using set theory to provide a model
of set theory. It seems a tad circular to me.

that one model of ZFC is as good as another?

As far as theorems are concerned, yes. If we wish to extend
ZFC by the addition of new axioms, we may have reason to prefer
some models over others.

In that case, by the completeness
theorem truth gets identified with provability and you become a
formalist.

Yes. The other option is to identify truth with a given model, which
is what you appear to doing, and urge the "correctness" or "naturalness"
of that model, or some such.

I may have you wrong here. Perhaps _you_ are the formalist. You
want to talk about sets as "just sets", i.e., the term taken as
primitive. What is true of them is what is provable in ZFC --
and that's all there is to it.

You can be a formalist if you want to. But it shouldn't
stop you from being able to do mathematics.

Right.

George's talk about "outer models" is just adding unnecessary
confusion to perfectly simple and easy-to-understand mathematics.

Not _that_ easy!

He's
wrong to say it's necessary to talk like this in order to clarify
something that needs to be clarified in my definition.

Actually, I understand George better than I understand you.
For what that's worth. Maybe we just share a similar philosophic
orientation on this issue. Maybe it's just that I'm a model
theory ignoramus.

My definition
is fine. The question of whether something goes through in ZFC is
machine-checkable and is independent of all these philosophical
scruples.

True.

Rhetorical question: What's your take on the continuum
hypothesis? Is it true in your intended, standard model?
See http://tinyurl.com/2h83x3 for Paul Cohen's view.
It's brief.

--
hz
.

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